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Bounded-hops power assignment in ad hoc wireless networks
Discrete Applied Mathematics 154 (2006) 1358 – 1371 www.elsevier.com/locate/dam
Bounded-hops power assignment in ad hoc wireless networks夡 G. Calinescu, S. Kapoor, M. Sarwat Department of Computer Science, Illinois Institute of Technology, Stuart Building, 10 West 31st Street, Chicago, IL 60616, USA Received 30 September 2003; received in revised form 4 August 2004; accepted 6 May 2005 Available online 20 January 2006
Abstract Motivated by topology control in ad hoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity). The input consists of a directed complete weighted digraph G = (V , c) (that is, c : V × V → R + ). The power of a vertex u in a directed spanning subgraph H is given by pH (u) = maxuv∈E(H ) c(uv), and corresponds to the energy consumption required for node u to transmit to all nodes v with uv ∈ E(H ). The power of H is given by p(H ) = u∈V pH (u). Power Assignment seeks to minimize p(H ) while H satisfies the given connectivity constraint. Min-Power Bounded-Hops Broadcast is a power assignment problem which has as additional input a positive integer d and a r ∈ V . The output H must be a r-rooted outgoing arborescence of depth at most d. We give an (O(log n), O(log n)) bicriteria approximation algorithm for Min-Power Bounded-Hops Broadcast: that is, our output has depth at most O(d log n) and power at most O(log n) times the optimum solution. For the Euclidean case, when c(u, v) = c(v, u) = u, v (here u, v is the Euclidean distance and is a constant between 2 and 5), the output of our algorithm can be modified to give a O((log n) ) approximation ratio. Previous results for Min-Power Bounded-Hops Broadcast are only exact algorithms based on dynamic programming for the case when the nodes lie on the line and c(u, v) = c(v, u) = u, v . Our bicriteria results extend to Min-Power Bounded-Hops Strong Connectivity, where H must have a path of at most d edges in between any two nodes. Previous work for Min-Power Bounded-Hops Strong Connectivity consists only of constant or better approximation for special cases of the Euclidean case. © 2005 Elsevier B.V. All rights reserved. Keywords: Approximation algorithms; Ad hoc wireless networks; Bicriteria approximation; Topology control
1. Introduction Wireless networking is an increasingly popular technology. Wireless devices utilize primarily batteries as a source of power and consequently efficient energy utilization is rapidly becoming an important issue. For the purpose of energy conservation, each node can (possibly dynamically) adjust its transmitting power, based on the distance to the receiving node and the background noise. Due to limited range, communication is achieved through intermediate nodes relaying packets. In several applications the quality of service also becomes an issue. For example, large delays may not be acceptable, and the number of total hops taken by a packet must be bounded. This motivates the study of the problems described below. 夡 A preliminary version of this paper appeared in Proceedings of the IEEE Wireless Communications and Networking Conference, pp. 2329–2334.
E-mail addresses: [email protected] (G. Calinescu), [email protected] (S. Kapoor), [email protected] (M. Sarwat). 0166-218X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2005.05.034
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In the most general case, a complete weighted directed graph H = (V , E) with non-negative power requirements c : E → R + is given by the positioning of the n wireless nodes, where c(u, v) represents the power requirement for the node u to establish a unidirectional link to node v. If the maximum range of u precludes reaching v in one hop, then c(uv) is set to ∞. Reflecting the broadcast nature of ad hoc wireless networks, once a node u transmits with power p(u), all nodes v with c(u, v) p(u) receive the signal. A function p : V → R + is called a power assignment, and it induces a directed graph, always denoted by G = (V , F ), with uv ∈ F whenever p(u)c(u, v). The goal of the Power Assignment problem is to minimize the total power v∈V p(v) such that the induced digraph satisfies a certain connectivity constraint. In this paper we consider three connectivity constraints: (1) Min-Power Bounded-Hops Broadcast, where the induced digraph must be an outgoing arborescence rooted at a given node r and of depth at most d. The vertex r and the integer d are given as a part of the input. (2) Min-Power Bounded-Hops Strong Connectivity, where the induced digraph must have a path of at most d edges in between any two nodes, where d is given as a part of the input. (3) Min-Power Bounded-Hops Symmetric Connectivity, where the symmetric restriction of the induced digraph must have diameter at most d, where d is given as a part of the input. The symmetric restriction of a directed graph is the undirected graph having an edge uv if and only if the digraph has both uv and vu. For simplicity of exposition, we use mostly the following equivalent definition of the Power Assignment problem: given a directed spanning subgraph H, define the power of a vertex u to be pH (u) = maxuv∈E(H ) c(uv) and the power of H as p(H ) = u∈V pH (u). To see the equivalence, note that an optimal power assignment inducing directed spanning subgraph H never has p(v) > maxuv∈E(H ) c(uv). Then the Power Assignment problem becomes finding the directed spanning subgraph H satisfying the connectivity constraint with minimum p(H ). An important special case (which we call the Euclidean case) is when the input graph G = (V , c) has power requirements given by c(u, v) = c(v, u) = u, v , where u, v is the Euclidean distance and is a constant between 2 and 5. This case is motivated by signal transmission in a network embedded in a two-dimensional space without any obstacles [25,26], with being the path-loss exponent. Minimizing the power contradicts bounding the number of hops in the induced subgraph, as it has been noted by [10,16] and formal tradeoff results in a similar but different model have been obtained by [23]. Indeed, if we look at the following example in the Euclidean case: (Figs. 1and 2) n points on the line with distance 1 in between two consecutive vertices, we note that without a bound on the number of hops Min-Power Broadcast (the version in which no restriction is put on the depth of the arborescence) has an optimum of n − 1, while with a bound of one hop, the unique solution requires (n − 1) power.
Root
Fig. 1. Unconstrained Min-Power broadcast.
Root
Fig. 2. Hop-constrained Min-Power broadcast.
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Numerous papers on Power Assignment have been published recently and we refer to [8] for a slightly out-of-date survey. We mention here only work on bounded-hops Power Assignment. Kirousis et al. [16] consider Min-Power Bounded-Hops Strong Connectivity when the power requirements are Euclidean and the nodes are equidistant on the line. In [7] Clementi et al. present a 2-approximation algorithm for Min-Power Bounded-Hops Strong Connectivity for the more general case when the nodes are on the line, but not equidistant. They also present better approximation bounds for the so-called well spread instances, when nodes’ positions satisfy that the ratio of the largest inter-node distance to the smallest inter-node distance is at most a constant times the square root of the number of nodes. In the Euclidean case, constant ratio algorithms for Min-Power Bounded-Hops Strong Connectivity for well spread instances have been obtained by [10]. The result of [7] is also implied by the exact algorithms obtained by [9] and [5] for Min-Power Bounded-Hops Broadcast when power requirement are Euclidean and the nodes are on the line. In a related work, Sanders et al. [2] discuss running time issues in calculating exactly bounded-hops min-power paths. Since this paper was submitted, two other related papers have been published. Among other results, Krumke et al. [19] obtain a bicriteria (O(log n), O(log n)) for Min-Power Bounded-Hops Symmetric Connectivity in the special case when the input graph has symmetric costs (that is, c(uv) = c(vu) for all nodes u, v). Ambuehl et al. [1] consider the Euclidean case and diameter d = 2. Our results do not make any assumptions regarding the input power requirement digraph—the algorithm can handle asymmetric power requirements, which are motivated by the possible existence of non-uniform wireless nodes and by applications in solving Network Lifetime [3]. However, our results use the bicriteria type of approximation introduced by Ravi et al. [27] and Marathe et al. [22]. In our case, bicriteria approximation allows for a relaxation of the constraint on the number of hops. The problem considered implicitly by [22] which is most related to ours is the Shallow Light Spanning Trees problem (SLST from now on): given an undirected graph with a specified vertex as root, find a minimum cost spanning tree with bounded radius, where the radius of the tree is the maximum length of a path from the root to a vertex of the graph, and length is a second given function unrelated to cost. We obtain (O(log n), O(log n)) bicriteria approximation algorithms for Min-Power Bounded-Hops Broadcast and Min-Power Bounded-Hops Strong Connectivity. That is, our output has number of hops bounded by O(d log n), and power at most O(log n) times the optimum solution with number of hops d. We present a straightforward reduction showing that improving our results for any of the problems we study in the case of arbitrary power requirements would lead to better ratios for SLST when the length function is 1 on all the edges, a rather hard problem. Since the publication of the conference version of [22] in 1995, the only progress for this restricted version of SLST has been reported by Kortsarz and Peleg [18]. As mentioned in [18], their results are also implied or improved by the Charikar et al. [6] algorithm for Directed Steiner Tree. Precisely, an immediate reduction followed by the algorithm of [6] gives a d log n or an O(n ) polynomial-time approximation for SLST (note that these are not bicriteria results). We note here that Min-Power Bounded-Hops Broadcast can also be immediately reduced to Directed Steiner Tree, similarly to the folklore reduction of Min-Power Broadcast which also appeared in [21]. For the Euclidean case, we can show that simple postprocessing gives an O((log n) )-approximation algorithm for Min-Power Bounded-Hops Broadcast. For Min-Power Bounded-Hops Strong Connectivity, the ratio we obtain is O((log n)+1 ) using a similar technique. Our methods are based on the bicriteria techniques of Marathe et al. [22], on the spider techniques of Klein and Ravi (used for Node-Weighted Steiner Tree [17]) as adapted for Power Assignment problems by [3], and on an extension of the Set Coverage problem [13]. Our results are centralized. If one is looking for fast (that is, with polylogarithmic number of rounds) distributed algorithms, it is a folklore result that the minimum power cannot be approximated better than n1/3− . The large number of distributed algorithms published in the networking literature either have instances using much more power than the optimum (such as [20,29]) or time (number of rounds) at least the diameter of the communication graph (this is the case with all the MST-based algorithms). The organization of the paper is as follows. Section 2 gives the main algorithm for Min-Power Bounded-Hops Broadcast, after introducing definitions and the reduction mentioned above. Section 3 discusses the extension of our techniques to Bounded-Hops Min-Power Strong Connectivity. The postprocessing technique for the Euclidean case is presented in Section 4. We present conclusions and open problems in Section 5.
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2. Min-power bounded-hops broadcast We start with a lemma which shows that Min-Power Bounded-Hops Broadcast problem is at least as hard to approximate as the conventional bicriteria problem of Shallow Light Spanning Tree. Therefore, any improvement of the guarantees presented here would imply a better than known guarantee for the SLST problem. Given the conventional bicriteria problem SLST we can reduce it to a Power Assignment problem by the following construction. Given an SLST instance G(V , E, c, d) with c(e) giving the cost of edge e and d being the bound on the diameter, construct a Min-Power Bounded-Hops Broadcast instance G = (V , E , c ) as follows: • V = V ∪ U , where U consists of two new vertices ve and ue for each edge of e = {u, v} ∈ E. • E = {(v, ve ), (ve , v), (ve , ue ), (ue , ve ), (ue , u), (u, ue ) | e ∈ E}. • c (v, ve ) = c (ve , v) = c (ue , u) = c (u, ue ) = 0 and c (ve , ue ) = c (ue , ve ) = c(e) for every e ∈ E. Let d = 3d + 1, and the root be the same vertex of V. 2.0.0.1 Claim 1. Given a tree T rooted at r in G such that all nodes are within d hops from r, there is a tree T rooted at r in G such that p(T ) = c(T ) and all nodes are within d = 3d + 1 hops of r. Proof. For every edge in e = (u, v) in T with u on the path from r to v in T, add to T the directed edges (u, ue ), (ue , ve ), and (ve , v). For every node u ∈ V , there is a path from r to u of no more than 3d hops. The only nodes not reached from r in T are in U. But since all these nodes are adjacent in G to a node in V, they can be reached from them in one hop at no additional cost. Thus we can construct a solution T from T which is of the same cost and radius at most 3d + 1. 2.0.0.2 Claim 2. Given a solution T in G with a diameter d , there is a solution T in G such that c(T )p(T ) and the radius of T is at most d /3. Proof. Modify the solution as follows: for every non-zero cost edge (ue , ve ) in T include e in T. Since every vertex of T has at most one non-zero cost edge incident to it, p(T ) = c(T ), and therefore c(T ) p(T ). Now let v ∈ V . T contains a directed path from r to v with at most d arcs, arcs which must come in triples. Each triple generates an edge in T, and these edges can be used for an undirected path of at most d /3 edges from r to v in T. Lemma 1. An algorithm with an approximation ratio of (O(f (n)), O(f (n))), for Min Power Bounded- Hops Broadcast implies a ratio of (O(f (n)), O(f (n))) for SLST. The proof is immediate from the above two claims. Given an instance I (d, n) of SLST with size n and diameter d, map it to an instance I (3d + 1, 3n) as illustrated in Claim 1, solve it with (O(f (3n)), O(f (3n))) ratio, and convert it back to a solution to I of the same cost and one third the diameter, implying an (O(f (n)), O(f (n))) guarantee. Now we present some definitions used by our algorithm for Min-Power Bounded-Hops Broadcast. By cost we mean power requirement c(uv). An approximation algorithm for the next problem will be used as a subroutine by our main algorithm. 2.0.0.3 Budgeted set coverage [15]. The input consists of a collection of sets R = {S1 , S2 , . . . , Sm } with associated prices {ci }m i=1 defined over a domain of elements X = {x1 , x2 , . . . , xn }. The goal is to find a collection of sets S ⊆ R, such that the total sum of the prices of the sets in S does not exceed the budget B, and the total number of elements covered is maximized. Khuller et al. [15] presented a (1 − 1/e)-approximation algorithm for Budgeted Set Coverage. However, their result does not apply directly since their algorithm examines all the sets and our instance has exponentially many sets.
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2.0.0.4 Branch. A branch is a directed graph S = (rS , VS , ES ) such that (VS , ES ) contains a directed path from the vertex rS (called the root of the branch) to every vertex of VS . A d-bounded branch is a branch such that every node is no more than d hops away from the root of the branch. The main algorithm assumes an estimate B on the power of the optimum solution is given. The algorithm finds a r-rooted spanning arborescence of depth O(d log n) and power O(B log n), if a solution of power at most B exists, but might terminate with a “failure” message if no such solution exists. A simple binary search on the value of B (in case of failure, the value of B is increased, otherwise a lower value is tried) would then give a bicriteria (O(log n), O(log n)) approximation for Min-Power Bounded-Hops Broadcast. 2.1. Main algorithm The main algorithm is an adaptation of the approximation algorithm of Marathe et al. [22] for the SLST problem. The algorithm works in phases and maintains a set of edges Q and a set P of nodes, called representatives, such that all the remaining nodes are reachable from P ∪ {r} by paths using edges in Q with small number of hops (the precise bound on the number of hops is given later in Lemma 5). In each phase, the algorithm computes a solution of a Budgeted Set Coverage instance attempting to reduce |P | by a constant factor without exceeding a given budget. The vertices of the branches found by the Budgeted Set Cover algorithm are removed from the set P and the root of each branch is added as a representative in P. Consequently |P | reduces. An illustration of one phase appears in Figs. 3 and 4. At the end of all the phases we obtain a subgraph (V , Q) from which we extract a shortest path arborescence where the length of the path is measured by the number of hops in the path. This is easily done via a breadth first search, and does not increase the power or the length in hops of the shortest path from r to other vertices of V. The detailed algorithm appears as Algorithm 1. u r
x
Fig. 3. The graph at the beginning of the phase. Solid circles denote representatives.
u r
x
Fig. 4. After the addition of two branchings (the dotted lines) only two representatives are left.
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Algorithm 1 Broadcast Require: A directed graph G(V , E, c), a diameter bound d, a budget B, and a root r. Ensure: Outputs a spanning arborescence A rooted at r, such that all the nodes are within O(d log |V |) hops of the root r and power(A) = O(B log |V |) provided a solution of power at most B exists; may exit with “failure” if no such solution exists. 1: P ← V \{r}, Q ← ∅ 2: While |P | 1 do 3: S ← setCoverage(P) 4: for all branches H (rH , VH , EH ) ∈ S do 5: Q ← Q ∪ EH 6: P ← P \VH 7: end for 8: for all H (rH , VH , EH ) ∈ S ∧ rH = r do 9: P ← P ∪ rH 10: end for 11: end while 12: Return the Breadth First Search Tree of (V , Q) 2.2. Solving the budgeted set coverage problem The Budgeted Set Coverage instance setCoverage(P ) used in Step 4 of the main algorithm Broadcast has as elements P, a set of vertices of G. The sets are given by certain d-bounded branches, the price of a branch is its power, and a branch Si covers a vertex v if v ∈ VSi . Precisely, we allow only valid branches, which are branches S with |VS ∩ P |2 or with rS = r. Such a restriction is needed to show a significant reduction (used later in the proof of Lemma 5) of |P |. There are exponentially many sets in this Budgeted Set Coverage problem, and this does not allow the direct application of the algorithm from [15]. We use a greedy ([14] and [13]) approach to the Budgeted Set Coverage problem. Given a collection of branches S, an element of P is called an uncovered node if it does not appear as a node in any of the branches. Assuming the optimum solution has power at most B, the procedure Greedy(U ) determines a set (a branch) with average power per covered node at most B/|U |, where U is the current set of uncovered nodes. Repeating this greedy choice produces a collection of branches which covers a good fraction (1/3) of the representatives. The detailed algorithm appears as Algorithm 2. Algorithm 2 setCoverage(P ) Require: A set of nodes P ⊆ V \{r}. Implicit input parameters are the directed graph G(V , E, c), diameter bound d, budget B, and root r. Ensure: Returns a set S of d-bounded valid branches such that their total power is no more than 2B and at least 1/3 of the nodes of P are covered, provided the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B; may exit with “failure” if no such solution exists. 1: C ← 0 2: U ← P 3: S ← 4: i ← 0 5: repeat 6: Si ← Greedy(U ) 7: U ← U \VSi 8: C ← C + p(Si )
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i ←i+1 S ← S ∪ {Si } until greedy returns failure or C B or U = ∅ if |U | > 23 |P | then exit failure else return S end if
2.3. Determining a good valid branch A valid branch S is good if and only if p(S)/|U ∩ VS | B/|U |, a property needed later in the proof of Lemma 3. Given a graph G, a set of uncovered nodes U and a vertex v, the goal is to find a good valid branch rooted at v, if such a branch exists. The algorithm builds W (as the tentative branch constructed by the algorithm) by first selecting a root v and a vertex w and setting W to consist of the arcs from v to those nodes u with c(vu)c(vw). Until W is a good valid branch or until W cannot be improved (in which case another root v or another vertex w is selected), the algorithm repeatedly and greedily adds to W shortest paths of appropriate number of hops from covered to uncovered nodes. The algorithm uses as a subroutine shortestBoundedPath(y, z, i) [12,28,24], which returns the path in G from vertex y to vertex z using at most i edges and having cost c minimum. The detailed algorithm is given in full below as Algorithm 3.
Algorithm 3 Greedy(U ) Require: Set of uncovered nodes U. Implicit input parameters are the directed graph G(V , E, c), diameter bound d, budget B, and root r. Ensure: Outputs a valid branch S such that p(S)/|VS ∩ U | B/|U |, provided the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B; may exit with “failure” if no such solution exists. 1: for all v ∈ V do 2: for all w ∈ V \{v} do 3: EW ← {vu | u = v ∧ c(vu) c(vw)} 4: Z ← VW ← {v} ∪ {u | c(vu) c(vw)} 5: while [p(W )/|U ∩ VW | > B/|U | or (|U ∩ VW | 1 and v = r)] and [U VW ] do 6: P ← miny∈Z,u∈U \VW shortestBoundedPath(y, u, d − 1) 7: EW ← EW ∪ EP , VW ← VW ∪ VP 8: end while 9: if p(W )/|U ∩ VW |B/|U | then 10: return (v, VW , EW ) 11: end if 12: end for 13: end for 14: return failure
2.4. Analysis and correctness The analysis proceeds in the reverse order, starting with the correctness of Algorithm Greedy and finishing with the bounds of Algorithm Broadcast. The next lemma is the counterpart of Lemma 4.1 and Theorem 3.1 of [17], and a variation of it is used in [3].
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Lemma 2. Assuming the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B, Algorithm Greedy finds a good valid branch. Proof. Let T be the optimum arborescence outgoing from the root (which has depth d, and we assume T has power at most B) and U be the set of uncovered nodes given as input to Greedy. Traverse T in postorder and whenever a vertex v is the ancestor of at least two vertices of U (where by default every vertex is an ancestor of itself) define a branch with root v given by the subtree of T rooted at v. Remove v and its descendants from T, and repeat. The process stops when |V (T ) ∩ U | < 2. If |V (T ) ∩ U | = 1, one last branch is given by the root r and the current T. Note that every obtained branch is valid: it either covers two vertices of U, or it has r as a root. Moreover every branch obtained has depth at most d. Let Si , for 1 i q, be the branches so obtained. Readers familiar with [17] will note that each branch Si is in fact a “spider”: the paths from the root to the vertices of outdegree zero are disjoint. It is immediate that p(S1 ) + p(S2 ) + · · · + p(Sq ) B. We have that |U ∩ VS1 | + |U ∩ VS2 | + · · · + |U ∩ VSq | = |U |. Therefore there exist S, a d-bounded branch (which is a spider) contained in T, with p(S)/|U ∩ VS | B/|U |, and either rS = r or |U ∩ VS | = 2. Let Q1 , Q2 , . . . , Qq , where q = |U ∩ VS |, be the paths of S from the children of rS (where some of these paths could have zero edges, and we use the convention that rS is also a child of itself) to vertices of U ∩ VS . For i = 1, 2, . . . , q, note that the paths Qi are edge disjoint and that Qi ∩ U consists of exactly one vertex, the last point of Qi . We denote by yi and ui the first vertex and last vertex of Qi . Moreover, assume p(Q1 ) p(Q2 ) · · · p(Qq ). Note that for a directed q path, its cost equals its power. Let wS be the vertex with c(rS wS ) = maxrS v∈E(S) c(rS v). Then p(S) = c(rS wS ) + i=1 p(Qi ). During the execution of Algorithm Greedy, the case v = rS and w = wS is considered and W (the tentative branch) is initialized to VW = {v} ∪ {u | c(vu) c(vw)} and EW = {vu | u = v ∧ c(vu)c(vw)}. Let k = |U ∩ {y | c(vy)c(vw)}|, where for convenience we use that c(vv) = 0. Let P1 , P2 , . . . , Pq−k be the first q − k results of the miny∈Z,u∈U \VW shortestBoundedPath(y, u, d − 1) procedure and note that each Pi contains only one vertex from U \W —its last vertex—as otherwise just a part of Pi would be returned by miny∈Z,u∈U \VW shortestBoundedPath (y, u, d − 1). For i =1, 2, . . . , q −k, we have that p(Pi )=c(Pi ) p(Qk+i )=c(Qk+i ). Indeed, before searching for the ith path, VW does not contain all of {u1 , u2 , . . . , uk+i }, while Z contains all of {y1 , y2 , . . . , yq }. Therefore, one of Q1 , Q2 , . . . , Qk+i is a candidate to be returned by miny∈Z,u∈U \VW shortestBoundedPath(y, u, d − 1), where we note that each Qi has at most d − 1 edges. If the algorithm does not find a good valid branch earlier, after q − k iterations of the while loop of q−k q Algorithm Greedy (with v = rS and w = wS ), W satisfies p(W ) c(vw) + i=1 p(Pi ) c(vw) + i=1 p(Qi ) = p(S) and |VW ∩ U | |VS ∩ U | and either |VW ∩ U |2 or v = r or both. We conclude that the algorithm always finds a good valid branch. The next lemma uses the method of Hochbaum and Pathria [14] (see also [13]) designed for analyzing the greedy algorithm for Set Coverage, the simpler version of Budgeted Set Coverage where all the sets have price 1. We did not attempt to optimize the two constants in the lemma. Lemma 3. Assuming the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B, the Algorithm setCoverage finds a set of valid branches of total power at most 2B and which covers at least |P |/3 nodes of P. Proof. The power of a branch returned by Algorithm Greedy can not be more than B (see Line 9 in the algorithm). And since the algorithm stops as soon as B is exceeded, the sum of the powers of the branches returned by Algorithm setCoverage is at most 2B. Now we show that at least |P |/3 nodes of P are covered. Let S1 , S2 , . . . , Sk be the branches returned by the Algorithm Greedy. Let U1 be P and Ui+1 be the set of uncovered nodes of P after the selection of branch Si by the greedy algorithm. Let qi = |Ui ∩ VSi |. Lemma 2 implies p(Si ) B B . = qi |Ui | |P | − i−1 j =1 qj
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Therefore
⎞ ⎛ i−1 p(Si ) ⎝ |P | − qj ⎠ qi B j =1
and after summing up these equations we obtain k−1 k k k i−1 k qj |P | |P | p(Si ) qi p(Si ) − qj = p(Si ) − B B B B i=1
i=1
Now we use k
i=1
k
i=1 p(Si ) B
qi |P | −
k−1
and
j =1
k
i=j +1 p(Si ) 2B
i=1
j =1
k
p(Si ).
i=j +1
to obtain
2qj
j =1
i=1
and therefore 3
k
qi |P |,
i=1
and we conclude that at least |P |/3 nodes of P are covered.
We continue with the proof of the correctness of the main algorithm Broadcast, which follows closely Marathe et al. [22]. Lemma 4. The number of phases of Algorithm Broadcast is O(log n). Proof. Let pi be the number of representatives before the ith phase, let qi be the number of representatives covered in the ith phase and ri be the number of roots of branches returned by the setCoverage procedure in phase i, where we exclude the branch rooted at r. Then the previous lemma gives qi pi /3, and since each branch except possibly the one rooted at r covers at least two representatives, we have ri qi /2. Therefore we have pi+1 pi − qi + ri pi − 21 qi 56 pi . Thus in each phase |P | decreases by a fraction of 1/6, and thus the number of phases is O(log n) as |P | is n − 1 to start with. The next two lemmas are the immediate counterpart of lemmas of [22] and we include their proofs for completeness. Lemma 5. The depth of the solution produced by Broadcast is O(d log n). Proof. It follows immediately by induction that after phase i, for every v ∈ V , Q contains a path of length at most d · i from some vertex of P to v. Lemma 6. The power of the output is O(B log n). Proof. The lemma follows immediately from the facts that there are O(log n) phases and in each we incur no more than 2B power. We now combine the above two lemmas to state the central result of this section:
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Theorem 7. Assuming the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B, the solution produced by Algorithm Broadcast has power O(B log n) and all the nodes are reachable from the root within O(d log n) hops. 3. Min-power bounded-hops strong connectivity In this section we use the result and methods of the previous section to give an approximation algorithm for the Min-Power Bounded-Hops Strong Connectivity problem with asymmetric power requirements. We again show how to solve a budgeted version of the problem. Let OPT be an optimal solution connecting all required nodes with paths of at most d hops and assume p(OPT) B. Let v be an arbitrary vertex. OPT contains an outgoing arborescence of depth at most d, called Aout , rooted at v (so p(Aout ) B) and an incoming arborescence of depth at most d, called Ain , rooted at v (so p(Ain ) OPT). Our algorithm also computes, and then puts together, an outgoing and an incoming arborescence rooted at v. The broadcast algorithm in the previous section produces an outgoing Qout , an arborescence of depth O(d log n) rooted at v with p(Qout ) O(B log n). To obtain an incoming arborescence Qin of depth O(d log n) and with p(Qin ) O(B log n) we use almost the same algorithm, but computing incoming branches instead of outgoing branches, and a modification described below. The main Algorithm Broadcast and the setCoverage algorithm are the same, except that incoming branches are used. Incoming branches have the same power as cost (each vertex has outdegree at most 1) and this makes finding good valid branches easier. An incoming arborescence can be also found by algorithms for SLST in directed graphs. Though we are not aware of published algorithms for SLST in directed graphs, the method of [22] can be modified to handle directed graphs (as we are dealing with spanning, and not Steiner trees). Our approach, which uses spiders and set coverage instead of matching (as in [22]) also works, and we describe it below for completeness’ sake. The modified Greedy procedure, which is described below as Algorithm 4, is simpler: we only consider minimum cost path of at most d hops from the uncovered nodes to the root of the branch. Algorithm 4 Modified_Greedy(U ) Require: Set of uncovered nodes U. Implicit input parameters are the directed graph G(V , E, c), diameter bound d, budget B, and root r. Ensure: Outputs a valid incoming branch S such that p(S)/|VS ∩U | B/|U |, provided the Min-Power BoundedHops Strong Connectivity instance has a solution with power at most B; may exit with “failure” if no such solution exists. 1: for all v ∈ V do 2: EW ← ∅ 3: VW ← {v} 4: while [p(W )/|U ∩ VW | > B/|U | or (|U ∩ VW | 1 and v = r)] and [U VW ] do 5: P ← minu∈U \VW shortestBoundedPath(u, v, d) 6: EW ← EW ∪ EP , VW ← VW ∪ VP 7: end while 8: if p(W )/|U ∩ VW |B/|U | then 9: return (v, VW , EW ) 10: end if 11: end for 12: return failure We proceed with translating Lemma 2, It uses the same proof idea and is simpler. Lemma 8. Assuming the Min-Power Bounded-Hops Strong Connectivity instance has a solution with power at most B, Algorithm 4 finds a good valid branch.
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Proof. Let T be the optimum arborescence incoming into the root (which has depth d, and we assume T has power at most B) and U be the set of uncovered nodes. Traverse T in postorder and whenever a vertex v is the ancestor of at least two vertices of U (where by default every vertex is an ancestor of itself) define a branch with root v given by the subtree of T rooted at v. Remove v and its descendants from T, and repeat. The process stops when |V (T ) ∩ U | < 2. If |V (T ) ∩ U | = 1, one last branch is given by the root r and the current T. Note that every obtained branch is valid: it either covers two vertices of U, or it has r as a root. Moreover every branch obtained has depth at most d. Let Si , for 1i q, be the branches so obtained. It is immediate that p(S1 ) + p(S2 ) + · · · + p(Sq ) B. We have that |U ∩ VS1 | + |U ∩ VS2 | + · · · + |U ∩ VSq | = |U |. Therefore there is d-bounded branch (which is a spider) of optimum S with p(S)/|U ∩ VS | B/|U |, and either rS = r or |U ∩ VS | = 2. Let Q1 , Q2 , . . . , Qq , where q = |U ∩ VS |, be the paths of S from the leafs S which are in U to rS (where we include the path with zero edges from rS to itself if rS ∈ U ). For i = 1, 2, . . . , q, note that the paths Qi are edge assume disjoint and that Qi ∩ U consists of exactly one vertex, the first point of Qi , which we denote by ui , Moreover, q p(Q1 ) p(Q2 ) · · · p(Qq ). Note that for a directed path, its cost equals its power. Then p(S) = i=1 p(Qi ). During the execution of Algorithm 4, the case v = rS is considered. Let k = 1 if v ∈ U ; otherwise let k = 0. Let P1 , P2 , . . . , Pq−k be the first q − k results of the minu∈U \VW shortestBoundedPath(u, v, d) procedure and note that each Pi contains only one vertex from U \W —its first vertex—as otherwise just a part of Pi would be returned by minu∈U \VW shortestBoundedPath(u, v, d). For i = 1, 2, . . . , q − k, we have that p(Pi ) = c(Pi ) p(Qk+i ) = c(Qk+i ). Indeed, before searching for the ith path, VW does not contain all of {u1 , u2 , . . . , uk+i }. Therefore, one of Q1 , Q2 , . . . , Qk+i is a candidate to be returned by minu∈U \VW shortestBoundedPath(u, v, d). where we note that each Qi has at most d edges. If the algorithm does not find a good valid branch earlier, after q − k iterations of the while loop of Algorithm 4 (with v = rS ), W satisfies q−k q p(W ) i=1 p(Pi ) i=1 p(Qi ) = p(S) and |VW ∩ U | |VS ∩ U | and either |VW ∩ U | 2 or v = r or both. We conclude that the algorithm always finds a good valid incoming branch. Lemmas 3–6 translate immediately, with exactly the same proofs. Since p(Qout ∪Qin ) p(Qout )+p(Qin ) O(B log n), and Qout ∪ Qin is a strongly connected spanning subgraph with diameter O(d log n), we have: Theorem 9. There is a polynomial-time algorithm for Min-Power Bounded-Hops Strong Connectivity whose output has power O(p(OPT) log n) and every node is reachable from another node by a path of length O(d log n), where OPT is an optimum solution of diameter d. 4. Approximation for min-power bounded-hops broadcast in the Euclidean case The approximation factors for the broadcast problem can be improved in the case when nodes represent points on a plane, and the cost c(uv) = c(v, u) = u, v , where u, v is the Euclidean distance and is a constant. The algorithm simply shortcuts paths of O(log n) hops (as illustrated in Fig. 5) and the pseudocode of the recursive procedure is given r
log n
log n q Fig. 5. Adding shortcuts to reduce depth.
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as Algorithm 5. The procedure is initially invoked with h = d. In the algorithm, q is such that the output of the Algorithm Broadcast has depth qd. Algorithm 5 Shortcut(T , h) Require: The input is an arborescence T rooted at r with depth h · q Ensure: The output T is an arborescence rooted at r, spanning V (T ), with depth h and p(T ) p(T )q −1 . if h = 0 then return end if L ← {u : depth(u) = q} M ← {u : depth(u) q} for all ui ∈ L do Ti ← subtree of T rooted at ui Ti ← Shortcut(Ti , h − 1) end for T ← ∪ui ∈L Ti ∪u∈M\{r} {ru}. return T Lemma 10. Algorithm 5 is correct, that is, T has depth h, and p(T ) p(T )q −1 . Proof. The depth property follows immediately by induction on h. We show that pT (r)q −1 u∈M\L pT (u) and the rest follows by induction. Let u be such that c(ru)=maxv∈M c(rv) and let P = x0 = r, x1 , . . . , xj = u be the directed path in T from r to u, and note that j q. Then ru j −1 j −1 i=0 xi xi+1 , as the worst case occurs when x0 x1 = x1 x2 = · · · = xj −1 xj . −1 Note that for i = 0, 1, . . . , j − 1 we have pT (xi ) xi xi+1 and that xi ∈ M\L. Therefore pT (r) q u∈M\L pT (u) and the lemma follows. We know by Lemmas 5 and 6 that we can assume that the output of the algorithm of Section 2 has depth d · q, with q = O(log n), and power O(p(OPT) log n), where OPT is an optimum solution of depth d. Based on the discussion above we have: Theorem 11. There is a O((log n) )-approximation algorithm for Min-Power Bounded-Hops Broadcast in the Euclidean case. The same postprocessing trick cannot be directly applied to Min-Power Bounded-Hops Strong Connectivity. For the outgoing arborescence (as in Section 3), the algorithm above for Broadcast is used. For the incoming arborescence, the method is described below. If d=O(log n), then the reduction to Directed Steiner Tree gives a O((log n)2 ) approximation. Else we let q = O(log n) be a parameter given by the upper bound on the number of phases of Algorithm 1 given in Lemma 4. Every bounded-depth path chosen by Algorithm 4 is compressed immediately by adding direct edges in between nodes q hops apart in the path. The power of the path is blown up by q · q −1 . Indeed, using the argument j of the broadcast case above, for any vertex u on the path, the new power of u satisfies p (u)q −1 i=1 p(ui ), where u is the ith vertex starting with u1 = u on the path from u to the root, and j q. Thus the total new power is i −1 j p(u ) q p(u), as each p(u) appears at most q times in the middle summation. i u p (u)q u i=1 There are O(log n) phases and in each phase the power used is bounded by O(Bq ) = O(Blog n), where B is the guess for the power of the optimum solution. Using 1, we conclude: Theorem 12. Min-Power Bounded-Hops Strong Connectivity in the Euclidean case admits a O((log n)+1 )approximation algorithm.
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5. Conclusions and open problems We provided (O(log n), O(log n)) bicriteria approximation algorithms for Min-Power Bounded-Hops Broadcast and Min-Power Bounded-Hops Strong Connectivity with asymmetric cost function. In the Euclidean case, postprocessing the output of the bicriteria Min-Power Bounded-Hops Broadcast leads to a O((log n) )-approximation algorithm. We believe our approach also works for Min-Power Bounded-Hops Symmetric Connectivity with asymmetric cost function (the Krumke et al. algorithm [19] relies on the cost function being symmetric), provided that shortest BoundedPath(y, z, i) and Algorithm 3 are modified appropriately. A symmetricBoundedPath(x, y, z, i) procedure, with four parameters (vertices x, y, z, and integer i) is used to find a minimum power directed graph P which contains the edge yx and a bidirected path from y to z using at most i edges. Such a directed graph P is found by applying the method described in [28,12,24] for bounded-hops minimum cost directed paths to a slight modification of the auxiliary directed graph whose construction is given in [4] for Min-Power Symmetric Unicast with asymmetric cost function. We can obtain a polynomial-time algorithm for Min-Power Bounded-Hops Symmetric Connectivity whose output has power O(p(OPT) log n) and diameter O(d log n), where OPT is an optimum solution of diameter d. We believe the equivalent of Theorem 12 also holds for Min-Power Bounded-Hops Symmetric Connectivity in the Euclidean case. The Euclidean case might be easier to solve. However, it would be reasonable to start with SLST first, before moving to minimizing power. As far as we know, nothing better than (O(log n), O(log n)) is known for SLST when hops are used for measuring diameter and simple Euclidean distance is used for measuring cost, except for the type of “jumps” we use in Section 4 to transform bicriteria results into pure approximation algorithms. We also note that with asymmetric costs, Min-Power Multicast (even with unbounded hops) is harder than Min-Power Broadcast: [3] presents a straightforward reduction from Directed Steiner Tree to Min-Power Multicast. The recent paper of Halperin and Krauthgamer [11] shows that Directed Steiner Tree does not admit any O(log2− n) approximation unless NP has quasi-polynomial Las Vegas algorithms. The techniques seem to easily generalize to bounded degree structures, just by considering bounded degree branches (or spiders) instead of bounded depth branches in the greedy algorithm of Section 2.3. However, in the Euclidean case there is no need to specially search for bounded-degree trees, as the minimum spanning tree is within a constant optimum for all three Min-Power problems, and can be assumed to have degree at most five. Acknowledgments The presentation of the paper benefitted from the careful reading of the referees. References [1] C. Ambuehl, A.E. Clementi, M.D. Ianni, N. Lev-Tov, A. Monti, D. Peleg, G. Rossi, R. Silvestri, Efficient algorithms for low-energy bounded-hop broadcast in ad hoc wireless networks, in: STACS, 2004. [2] P. Beir, P. Sanders, N. Sivadasan, Energy optimal routing in radio networks using geometric data structrures, ICALP, Lecture notes in Computer Science, vol. 2380, 2002, pp. 366–376. [3] G. Calinescu, S. Kapoor, A. Olshevsky, A. Zelikovsky, Network lifetime and power assignment in ad hoc wireless networks, in: Proceedings of the 11th European Symposium on Algorithms, 2003. [4] G. Calinescu, I. Mandoiu, A. Zelikovsky, Symmetric connectivity with minimum power consumption in radio networks, in: Second IFIP International Conference on Theoretical Computer Science, 2002, pp. 119–130. [5] I. Caragiannis, C. Kaklamanis, P. Kanellopoulos, New results for energy-efficient broadcasting in wireless networks, in: ISAAC’ 2002, 2002. [6] M. Charikar, C. Chekuri, T. Cheung, Z. Dai, A. Goel, S. Guha, M. Li, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1) (1999) 73–91. [7] A. Clementi, A. Ferreira, P. Penna, S. Perennes, R. Silvestri, The minimum range assignment problem on linear radio networks, ESA ‘00, Lecture Notes in Computer Science, vol. 1879, 2000, pp. 143–154. [8] A. Clementi, G. Huiban, P. Penna, G. Rossi, Y. Verhoeven, Some recent theoretical advances and open questions on energy consumption in ad hoc wireless networks, in: ARACNE, 2002. [9] A.E. Clementi, M.D. Ianni, R. Silvestri, The minimum broadcast range assignment problem on linear multi-hop networks, Theoret. Comput. Sci. 1-3 (299) (2003) 751–761. [10] A.E. Clementi, P. Penna, R. Silvestri, On the power assignment problem in radio networks, Electronic Colloquium on Computational Complexity Report TR00-054. [11] E. Halperin, R. Krauthgamer, Polylogarithmic inapproximability, in: Proceedings of the 35th ACM Symposium on Theory of Computing, 2003, pp. 585–594.
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[12] R. Hassin, Approximation schemes for the restricted shortest path problem, Math. Oper. Res. 17 (1) (1992) 36–42. [13] D.S. Hochbaum, Approximating covering and packing problems, in: D.S. Hochbaum (Ed.), Approximation Algorithms for NP-Hard Problems, PWS Publishing Co., Boston, MA, 1996. [14] D. Hochbaum, A. Pathria, Analysis of the greedy approach in covering problems, unpublished manuscript, 1994. [15] S. Khuller, A. Moss, J. Naor, The budgeted maximum coverage problem, Inform. Process. Lett. 70 (1) (1999) 39–45. [16] L.M. Kirousis, E. Kranakis, D. Krizanc, A. Pelc, Power consumption in packet radio networks, Theoret. Comput. Sci. 243 (2000) 289–305. [17] P. Klein, R. Ravi, A nearly best-possible approximation algorithm for node-weighted Steiner trees, J. Algorithms 19 (1995) 104–115. [18] G. Kortsarz, D. Peleg, Approximating the weight of shallow Steiner trees, DAMATH: Discrete Applied Mathematics and Combinatorial Operations Research and Computer Science 93. [19] S. Krumke, R. Liu, E. Lloyd, M. Marathe, R. Ramanathan, S. Ravi, Topology control problems under symmetric and asymmetric power thresholds, in: Proc. Ad Hoc Now, 2003. [20] N. Li, J. Hou, L. Sha, Design and analysis of an MST-based topology control algorithm, in: IEEE INFOCOM, 2003. [21] W. Liang, Constructing minimum-energy broadcast trees in wireless ad hoc networks, in: Proceedings of the Third ACM International Symposium on Mobile Ad Hoc Networking & Computing, ACM Press, New York, 2002, pp. 112–122. [22] M. Marathe, R. Ravi, R. Sundaram, S. Ravi, D. Rosenkrantz H.H. III, Bicriteria network design problems, J. Algorithms 28 (1998) 142–171. [23] F. Meyer auf der Heide, C. Schindelhauer, K. Volbert, M. Grünewald, Congestion, dilation, and energy in radio networks, Theory Comput. Systems 37 (3) (2004) 343–370. [24] C. Phillips, The network inhibition problem, in: STOC, 1993, pp. 776–785. [25] R. Rajaraman, Topology control and routing in ad hoc networks: a survey, SIGACT News 33 (2002) 60–73. [26] T. Rappaport, Wireless Communications: Principles and Practices, Prentice-Hall, Englewood Cliffs, NJ, 1996. [27] R. Ravi, M.V. Marathe, S.S. Ravi, D.J. Rosenkrantz, H.B. Hunt, Approximation algorithms for degree constrained minimum cost network-design problems, Algorithmica 31 (1) (2001) 58–78. [28] A. Warburton, Approximation of Pareto optima in multiple-objective, shortest path problems, Oper. Res. 35 (1987) 70–79. [29] R. Wattenhofer, L. Li, P. Bahl, Y.-M. Wang, Distributed topology control for wireless multihop ad hoc networks, in: IEEE INFOCOM’01, 2001.
Bounded-hops power assignment in ad hoc wireless networks夡 G. Calinescu, S. Kapoor, M. Sarwat Department of Computer Science, Illinois Institute of Technology, Stuart Building, 10 West 31st Street, Chicago, IL 60616, USA Received 30 September 2003; received in revised form 4 August 2004; accepted 6 May 2005 Available online 20 January 2006
Abstract Motivated by topology control in ad hoc wireless networks, Power Assignment is a family of problems, each defined by a certain connectivity constraint (such as strong connectivity). The input consists of a directed complete weighted digraph G = (V , c) (that is, c : V × V → R + ). The power of a vertex u in a directed spanning subgraph H is given by pH (u) = maxuv∈E(H ) c(uv), and corresponds to the energy consumption required for node u to transmit to all nodes v with uv ∈ E(H ). The power of H is given by p(H ) = u∈V pH (u). Power Assignment seeks to minimize p(H ) while H satisfies the given connectivity constraint. Min-Power Bounded-Hops Broadcast is a power assignment problem which has as additional input a positive integer d and a r ∈ V . The output H must be a r-rooted outgoing arborescence of depth at most d. We give an (O(log n), O(log n)) bicriteria approximation algorithm for Min-Power Bounded-Hops Broadcast: that is, our output has depth at most O(d log n) and power at most O(log n) times the optimum solution. For the Euclidean case, when c(u, v) = c(v, u) = u, v (here u, v is the Euclidean distance and is a constant between 2 and 5), the output of our algorithm can be modified to give a O((log n) ) approximation ratio. Previous results for Min-Power Bounded-Hops Broadcast are only exact algorithms based on dynamic programming for the case when the nodes lie on the line and c(u, v) = c(v, u) = u, v . Our bicriteria results extend to Min-Power Bounded-Hops Strong Connectivity, where H must have a path of at most d edges in between any two nodes. Previous work for Min-Power Bounded-Hops Strong Connectivity consists only of constant or better approximation for special cases of the Euclidean case. © 2005 Elsevier B.V. All rights reserved. Keywords: Approximation algorithms; Ad hoc wireless networks; Bicriteria approximation; Topology control
1. Introduction Wireless networking is an increasingly popular technology. Wireless devices utilize primarily batteries as a source of power and consequently efficient energy utilization is rapidly becoming an important issue. For the purpose of energy conservation, each node can (possibly dynamically) adjust its transmitting power, based on the distance to the receiving node and the background noise. Due to limited range, communication is achieved through intermediate nodes relaying packets. In several applications the quality of service also becomes an issue. For example, large delays may not be acceptable, and the number of total hops taken by a packet must be bounded. This motivates the study of the problems described below. 夡 A preliminary version of this paper appeared in Proceedings of the IEEE Wireless Communications and Networking Conference, pp. 2329–2334.
E-mail addresses: [email protected] (G. Calinescu), [email protected] (S. Kapoor), [email protected] (M. Sarwat). 0166-218X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.dam.2005.05.034
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In the most general case, a complete weighted directed graph H = (V , E) with non-negative power requirements c : E → R + is given by the positioning of the n wireless nodes, where c(u, v) represents the power requirement for the node u to establish a unidirectional link to node v. If the maximum range of u precludes reaching v in one hop, then c(uv) is set to ∞. Reflecting the broadcast nature of ad hoc wireless networks, once a node u transmits with power p(u), all nodes v with c(u, v) p(u) receive the signal. A function p : V → R + is called a power assignment, and it induces a directed graph, always denoted by G = (V , F ), with uv ∈ F whenever p(u)c(u, v). The goal of the Power Assignment problem is to minimize the total power v∈V p(v) such that the induced digraph satisfies a certain connectivity constraint. In this paper we consider three connectivity constraints: (1) Min-Power Bounded-Hops Broadcast, where the induced digraph must be an outgoing arborescence rooted at a given node r and of depth at most d. The vertex r and the integer d are given as a part of the input. (2) Min-Power Bounded-Hops Strong Connectivity, where the induced digraph must have a path of at most d edges in between any two nodes, where d is given as a part of the input. (3) Min-Power Bounded-Hops Symmetric Connectivity, where the symmetric restriction of the induced digraph must have diameter at most d, where d is given as a part of the input. The symmetric restriction of a directed graph is the undirected graph having an edge uv if and only if the digraph has both uv and vu. For simplicity of exposition, we use mostly the following equivalent definition of the Power Assignment problem: given a directed spanning subgraph H, define the power of a vertex u to be pH (u) = maxuv∈E(H ) c(uv) and the power of H as p(H ) = u∈V pH (u). To see the equivalence, note that an optimal power assignment inducing directed spanning subgraph H never has p(v) > maxuv∈E(H ) c(uv). Then the Power Assignment problem becomes finding the directed spanning subgraph H satisfying the connectivity constraint with minimum p(H ). An important special case (which we call the Euclidean case) is when the input graph G = (V , c) has power requirements given by c(u, v) = c(v, u) = u, v , where u, v is the Euclidean distance and is a constant between 2 and 5. This case is motivated by signal transmission in a network embedded in a two-dimensional space without any obstacles [25,26], with being the path-loss exponent. Minimizing the power contradicts bounding the number of hops in the induced subgraph, as it has been noted by [10,16] and formal tradeoff results in a similar but different model have been obtained by [23]. Indeed, if we look at the following example in the Euclidean case: (Figs. 1and 2) n points on the line with distance 1 in between two consecutive vertices, we note that without a bound on the number of hops Min-Power Broadcast (the version in which no restriction is put on the depth of the arborescence) has an optimum of n − 1, while with a bound of one hop, the unique solution requires (n − 1) power.
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Fig. 1. Unconstrained Min-Power broadcast.
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Fig. 2. Hop-constrained Min-Power broadcast.
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Numerous papers on Power Assignment have been published recently and we refer to [8] for a slightly out-of-date survey. We mention here only work on bounded-hops Power Assignment. Kirousis et al. [16] consider Min-Power Bounded-Hops Strong Connectivity when the power requirements are Euclidean and the nodes are equidistant on the line. In [7] Clementi et al. present a 2-approximation algorithm for Min-Power Bounded-Hops Strong Connectivity for the more general case when the nodes are on the line, but not equidistant. They also present better approximation bounds for the so-called well spread instances, when nodes’ positions satisfy that the ratio of the largest inter-node distance to the smallest inter-node distance is at most a constant times the square root of the number of nodes. In the Euclidean case, constant ratio algorithms for Min-Power Bounded-Hops Strong Connectivity for well spread instances have been obtained by [10]. The result of [7] is also implied by the exact algorithms obtained by [9] and [5] for Min-Power Bounded-Hops Broadcast when power requirement are Euclidean and the nodes are on the line. In a related work, Sanders et al. [2] discuss running time issues in calculating exactly bounded-hops min-power paths. Since this paper was submitted, two other related papers have been published. Among other results, Krumke et al. [19] obtain a bicriteria (O(log n), O(log n)) for Min-Power Bounded-Hops Symmetric Connectivity in the special case when the input graph has symmetric costs (that is, c(uv) = c(vu) for all nodes u, v). Ambuehl et al. [1] consider the Euclidean case and diameter d = 2. Our results do not make any assumptions regarding the input power requirement digraph—the algorithm can handle asymmetric power requirements, which are motivated by the possible existence of non-uniform wireless nodes and by applications in solving Network Lifetime [3]. However, our results use the bicriteria type of approximation introduced by Ravi et al. [27] and Marathe et al. [22]. In our case, bicriteria approximation allows for a relaxation of the constraint on the number of hops. The problem considered implicitly by [22] which is most related to ours is the Shallow Light Spanning Trees problem (SLST from now on): given an undirected graph with a specified vertex as root, find a minimum cost spanning tree with bounded radius, where the radius of the tree is the maximum length of a path from the root to a vertex of the graph, and length is a second given function unrelated to cost. We obtain (O(log n), O(log n)) bicriteria approximation algorithms for Min-Power Bounded-Hops Broadcast and Min-Power Bounded-Hops Strong Connectivity. That is, our output has number of hops bounded by O(d log n), and power at most O(log n) times the optimum solution with number of hops d. We present a straightforward reduction showing that improving our results for any of the problems we study in the case of arbitrary power requirements would lead to better ratios for SLST when the length function is 1 on all the edges, a rather hard problem. Since the publication of the conference version of [22] in 1995, the only progress for this restricted version of SLST has been reported by Kortsarz and Peleg [18]. As mentioned in [18], their results are also implied or improved by the Charikar et al. [6] algorithm for Directed Steiner Tree. Precisely, an immediate reduction followed by the algorithm of [6] gives a d log n or an O(n ) polynomial-time approximation for SLST (note that these are not bicriteria results). We note here that Min-Power Bounded-Hops Broadcast can also be immediately reduced to Directed Steiner Tree, similarly to the folklore reduction of Min-Power Broadcast which also appeared in [21]. For the Euclidean case, we can show that simple postprocessing gives an O((log n) )-approximation algorithm for Min-Power Bounded-Hops Broadcast. For Min-Power Bounded-Hops Strong Connectivity, the ratio we obtain is O((log n)+1 ) using a similar technique. Our methods are based on the bicriteria techniques of Marathe et al. [22], on the spider techniques of Klein and Ravi (used for Node-Weighted Steiner Tree [17]) as adapted for Power Assignment problems by [3], and on an extension of the Set Coverage problem [13]. Our results are centralized. If one is looking for fast (that is, with polylogarithmic number of rounds) distributed algorithms, it is a folklore result that the minimum power cannot be approximated better than n1/3− . The large number of distributed algorithms published in the networking literature either have instances using much more power than the optimum (such as [20,29]) or time (number of rounds) at least the diameter of the communication graph (this is the case with all the MST-based algorithms). The organization of the paper is as follows. Section 2 gives the main algorithm for Min-Power Bounded-Hops Broadcast, after introducing definitions and the reduction mentioned above. Section 3 discusses the extension of our techniques to Bounded-Hops Min-Power Strong Connectivity. The postprocessing technique for the Euclidean case is presented in Section 4. We present conclusions and open problems in Section 5.
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2. Min-power bounded-hops broadcast We start with a lemma which shows that Min-Power Bounded-Hops Broadcast problem is at least as hard to approximate as the conventional bicriteria problem of Shallow Light Spanning Tree. Therefore, any improvement of the guarantees presented here would imply a better than known guarantee for the SLST problem. Given the conventional bicriteria problem SLST we can reduce it to a Power Assignment problem by the following construction. Given an SLST instance G(V , E, c, d) with c(e) giving the cost of edge e and d being the bound on the diameter, construct a Min-Power Bounded-Hops Broadcast instance G = (V , E , c ) as follows: • V = V ∪ U , where U consists of two new vertices ve and ue for each edge of e = {u, v} ∈ E. • E = {(v, ve ), (ve , v), (ve , ue ), (ue , ve ), (ue , u), (u, ue ) | e ∈ E}. • c (v, ve ) = c (ve , v) = c (ue , u) = c (u, ue ) = 0 and c (ve , ue ) = c (ue , ve ) = c(e) for every e ∈ E. Let d = 3d + 1, and the root be the same vertex of V. 2.0.0.1 Claim 1. Given a tree T rooted at r in G such that all nodes are within d hops from r, there is a tree T rooted at r in G such that p(T ) = c(T ) and all nodes are within d = 3d + 1 hops of r. Proof. For every edge in e = (u, v) in T with u on the path from r to v in T, add to T the directed edges (u, ue ), (ue , ve ), and (ve , v). For every node u ∈ V , there is a path from r to u of no more than 3d hops. The only nodes not reached from r in T are in U. But since all these nodes are adjacent in G to a node in V, they can be reached from them in one hop at no additional cost. Thus we can construct a solution T from T which is of the same cost and radius at most 3d + 1. 2.0.0.2 Claim 2. Given a solution T in G with a diameter d , there is a solution T in G such that c(T )p(T ) and the radius of T is at most d /3. Proof. Modify the solution as follows: for every non-zero cost edge (ue , ve ) in T include e in T. Since every vertex of T has at most one non-zero cost edge incident to it, p(T ) = c(T ), and therefore c(T ) p(T ). Now let v ∈ V . T contains a directed path from r to v with at most d arcs, arcs which must come in triples. Each triple generates an edge in T, and these edges can be used for an undirected path of at most d /3 edges from r to v in T. Lemma 1. An algorithm with an approximation ratio of (O(f (n)), O(f (n))), for Min Power Bounded- Hops Broadcast implies a ratio of (O(f (n)), O(f (n))) for SLST. The proof is immediate from the above two claims. Given an instance I (d, n) of SLST with size n and diameter d, map it to an instance I (3d + 1, 3n) as illustrated in Claim 1, solve it with (O(f (3n)), O(f (3n))) ratio, and convert it back to a solution to I of the same cost and one third the diameter, implying an (O(f (n)), O(f (n))) guarantee. Now we present some definitions used by our algorithm for Min-Power Bounded-Hops Broadcast. By cost we mean power requirement c(uv). An approximation algorithm for the next problem will be used as a subroutine by our main algorithm. 2.0.0.3 Budgeted set coverage [15]. The input consists of a collection of sets R = {S1 , S2 , . . . , Sm } with associated prices {ci }m i=1 defined over a domain of elements X = {x1 , x2 , . . . , xn }. The goal is to find a collection of sets S ⊆ R, such that the total sum of the prices of the sets in S does not exceed the budget B, and the total number of elements covered is maximized. Khuller et al. [15] presented a (1 − 1/e)-approximation algorithm for Budgeted Set Coverage. However, their result does not apply directly since their algorithm examines all the sets and our instance has exponentially many sets.
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2.0.0.4 Branch. A branch is a directed graph S = (rS , VS , ES ) such that (VS , ES ) contains a directed path from the vertex rS (called the root of the branch) to every vertex of VS . A d-bounded branch is a branch such that every node is no more than d hops away from the root of the branch. The main algorithm assumes an estimate B on the power of the optimum solution is given. The algorithm finds a r-rooted spanning arborescence of depth O(d log n) and power O(B log n), if a solution of power at most B exists, but might terminate with a “failure” message if no such solution exists. A simple binary search on the value of B (in case of failure, the value of B is increased, otherwise a lower value is tried) would then give a bicriteria (O(log n), O(log n)) approximation for Min-Power Bounded-Hops Broadcast. 2.1. Main algorithm The main algorithm is an adaptation of the approximation algorithm of Marathe et al. [22] for the SLST problem. The algorithm works in phases and maintains a set of edges Q and a set P of nodes, called representatives, such that all the remaining nodes are reachable from P ∪ {r} by paths using edges in Q with small number of hops (the precise bound on the number of hops is given later in Lemma 5). In each phase, the algorithm computes a solution of a Budgeted Set Coverage instance attempting to reduce |P | by a constant factor without exceeding a given budget. The vertices of the branches found by the Budgeted Set Cover algorithm are removed from the set P and the root of each branch is added as a representative in P. Consequently |P | reduces. An illustration of one phase appears in Figs. 3 and 4. At the end of all the phases we obtain a subgraph (V , Q) from which we extract a shortest path arborescence where the length of the path is measured by the number of hops in the path. This is easily done via a breadth first search, and does not increase the power or the length in hops of the shortest path from r to other vertices of V. The detailed algorithm appears as Algorithm 1. u r
x
Fig. 3. The graph at the beginning of the phase. Solid circles denote representatives.
u r
x
Fig. 4. After the addition of two branchings (the dotted lines) only two representatives are left.
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Algorithm 1 Broadcast Require: A directed graph G(V , E, c), a diameter bound d, a budget B, and a root r. Ensure: Outputs a spanning arborescence A rooted at r, such that all the nodes are within O(d log |V |) hops of the root r and power(A) = O(B log |V |) provided a solution of power at most B exists; may exit with “failure” if no such solution exists. 1: P ← V \{r}, Q ← ∅ 2: While |P | 1 do 3: S ← setCoverage(P) 4: for all branches H (rH , VH , EH ) ∈ S do 5: Q ← Q ∪ EH 6: P ← P \VH 7: end for 8: for all H (rH , VH , EH ) ∈ S ∧ rH = r do 9: P ← P ∪ rH 10: end for 11: end while 12: Return the Breadth First Search Tree of (V , Q) 2.2. Solving the budgeted set coverage problem The Budgeted Set Coverage instance setCoverage(P ) used in Step 4 of the main algorithm Broadcast has as elements P, a set of vertices of G. The sets are given by certain d-bounded branches, the price of a branch is its power, and a branch Si covers a vertex v if v ∈ VSi . Precisely, we allow only valid branches, which are branches S with |VS ∩ P |2 or with rS = r. Such a restriction is needed to show a significant reduction (used later in the proof of Lemma 5) of |P |. There are exponentially many sets in this Budgeted Set Coverage problem, and this does not allow the direct application of the algorithm from [15]. We use a greedy ([14] and [13]) approach to the Budgeted Set Coverage problem. Given a collection of branches S, an element of P is called an uncovered node if it does not appear as a node in any of the branches. Assuming the optimum solution has power at most B, the procedure Greedy(U ) determines a set (a branch) with average power per covered node at most B/|U |, where U is the current set of uncovered nodes. Repeating this greedy choice produces a collection of branches which covers a good fraction (1/3) of the representatives. The detailed algorithm appears as Algorithm 2. Algorithm 2 setCoverage(P ) Require: A set of nodes P ⊆ V \{r}. Implicit input parameters are the directed graph G(V , E, c), diameter bound d, budget B, and root r. Ensure: Returns a set S of d-bounded valid branches such that their total power is no more than 2B and at least 1/3 of the nodes of P are covered, provided the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B; may exit with “failure” if no such solution exists. 1: C ← 0 2: U ← P 3: S ← 4: i ← 0 5: repeat 6: Si ← Greedy(U ) 7: U ← U \VSi 8: C ← C + p(Si )
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i ←i+1 S ← S ∪ {Si } until greedy returns failure or C B or U = ∅ if |U | > 23 |P | then exit failure else return S end if
2.3. Determining a good valid branch A valid branch S is good if and only if p(S)/|U ∩ VS | B/|U |, a property needed later in the proof of Lemma 3. Given a graph G, a set of uncovered nodes U and a vertex v, the goal is to find a good valid branch rooted at v, if such a branch exists. The algorithm builds W (as the tentative branch constructed by the algorithm) by first selecting a root v and a vertex w and setting W to consist of the arcs from v to those nodes u with c(vu)c(vw). Until W is a good valid branch or until W cannot be improved (in which case another root v or another vertex w is selected), the algorithm repeatedly and greedily adds to W shortest paths of appropriate number of hops from covered to uncovered nodes. The algorithm uses as a subroutine shortestBoundedPath(y, z, i) [12,28,24], which returns the path in G from vertex y to vertex z using at most i edges and having cost c minimum. The detailed algorithm is given in full below as Algorithm 3.
Algorithm 3 Greedy(U ) Require: Set of uncovered nodes U. Implicit input parameters are the directed graph G(V , E, c), diameter bound d, budget B, and root r. Ensure: Outputs a valid branch S such that p(S)/|VS ∩ U | B/|U |, provided the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B; may exit with “failure” if no such solution exists. 1: for all v ∈ V do 2: for all w ∈ V \{v} do 3: EW ← {vu | u = v ∧ c(vu) c(vw)} 4: Z ← VW ← {v} ∪ {u | c(vu) c(vw)} 5: while [p(W )/|U ∩ VW | > B/|U | or (|U ∩ VW | 1 and v = r)] and [U VW ] do 6: P ← miny∈Z,u∈U \VW shortestBoundedPath(y, u, d − 1) 7: EW ← EW ∪ EP , VW ← VW ∪ VP 8: end while 9: if p(W )/|U ∩ VW |B/|U | then 10: return (v, VW , EW ) 11: end if 12: end for 13: end for 14: return failure
2.4. Analysis and correctness The analysis proceeds in the reverse order, starting with the correctness of Algorithm Greedy and finishing with the bounds of Algorithm Broadcast. The next lemma is the counterpart of Lemma 4.1 and Theorem 3.1 of [17], and a variation of it is used in [3].
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Lemma 2. Assuming the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B, Algorithm Greedy finds a good valid branch. Proof. Let T be the optimum arborescence outgoing from the root (which has depth d, and we assume T has power at most B) and U be the set of uncovered nodes given as input to Greedy. Traverse T in postorder and whenever a vertex v is the ancestor of at least two vertices of U (where by default every vertex is an ancestor of itself) define a branch with root v given by the subtree of T rooted at v. Remove v and its descendants from T, and repeat. The process stops when |V (T ) ∩ U | < 2. If |V (T ) ∩ U | = 1, one last branch is given by the root r and the current T. Note that every obtained branch is valid: it either covers two vertices of U, or it has r as a root. Moreover every branch obtained has depth at most d. Let Si , for 1 i q, be the branches so obtained. Readers familiar with [17] will note that each branch Si is in fact a “spider”: the paths from the root to the vertices of outdegree zero are disjoint. It is immediate that p(S1 ) + p(S2 ) + · · · + p(Sq ) B. We have that |U ∩ VS1 | + |U ∩ VS2 | + · · · + |U ∩ VSq | = |U |. Therefore there exist S, a d-bounded branch (which is a spider) contained in T, with p(S)/|U ∩ VS | B/|U |, and either rS = r or |U ∩ VS | = 2. Let Q1 , Q2 , . . . , Qq , where q = |U ∩ VS |, be the paths of S from the children of rS (where some of these paths could have zero edges, and we use the convention that rS is also a child of itself) to vertices of U ∩ VS . For i = 1, 2, . . . , q, note that the paths Qi are edge disjoint and that Qi ∩ U consists of exactly one vertex, the last point of Qi . We denote by yi and ui the first vertex and last vertex of Qi . Moreover, assume p(Q1 ) p(Q2 ) · · · p(Qq ). Note that for a directed q path, its cost equals its power. Let wS be the vertex with c(rS wS ) = maxrS v∈E(S) c(rS v). Then p(S) = c(rS wS ) + i=1 p(Qi ). During the execution of Algorithm Greedy, the case v = rS and w = wS is considered and W (the tentative branch) is initialized to VW = {v} ∪ {u | c(vu) c(vw)} and EW = {vu | u = v ∧ c(vu)c(vw)}. Let k = |U ∩ {y | c(vy)c(vw)}|, where for convenience we use that c(vv) = 0. Let P1 , P2 , . . . , Pq−k be the first q − k results of the miny∈Z,u∈U \VW shortestBoundedPath(y, u, d − 1) procedure and note that each Pi contains only one vertex from U \W —its last vertex—as otherwise just a part of Pi would be returned by miny∈Z,u∈U \VW shortestBoundedPath (y, u, d − 1). For i =1, 2, . . . , q −k, we have that p(Pi )=c(Pi ) p(Qk+i )=c(Qk+i ). Indeed, before searching for the ith path, VW does not contain all of {u1 , u2 , . . . , uk+i }, while Z contains all of {y1 , y2 , . . . , yq }. Therefore, one of Q1 , Q2 , . . . , Qk+i is a candidate to be returned by miny∈Z,u∈U \VW shortestBoundedPath(y, u, d − 1), where we note that each Qi has at most d − 1 edges. If the algorithm does not find a good valid branch earlier, after q − k iterations of the while loop of q−k q Algorithm Greedy (with v = rS and w = wS ), W satisfies p(W ) c(vw) + i=1 p(Pi ) c(vw) + i=1 p(Qi ) = p(S) and |VW ∩ U | |VS ∩ U | and either |VW ∩ U |2 or v = r or both. We conclude that the algorithm always finds a good valid branch. The next lemma uses the method of Hochbaum and Pathria [14] (see also [13]) designed for analyzing the greedy algorithm for Set Coverage, the simpler version of Budgeted Set Coverage where all the sets have price 1. We did not attempt to optimize the two constants in the lemma. Lemma 3. Assuming the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B, the Algorithm setCoverage finds a set of valid branches of total power at most 2B and which covers at least |P |/3 nodes of P. Proof. The power of a branch returned by Algorithm Greedy can not be more than B (see Line 9 in the algorithm). And since the algorithm stops as soon as B is exceeded, the sum of the powers of the branches returned by Algorithm setCoverage is at most 2B. Now we show that at least |P |/3 nodes of P are covered. Let S1 , S2 , . . . , Sk be the branches returned by the Algorithm Greedy. Let U1 be P and Ui+1 be the set of uncovered nodes of P after the selection of branch Si by the greedy algorithm. Let qi = |Ui ∩ VSi |. Lemma 2 implies p(Si ) B B . = qi |Ui | |P | − i−1 j =1 qj
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Therefore
⎞ ⎛ i−1 p(Si ) ⎝ |P | − qj ⎠ qi B j =1
and after summing up these equations we obtain k−1 k k k i−1 k qj |P | |P | p(Si ) qi p(Si ) − qj = p(Si ) − B B B B i=1
i=1
Now we use k
i=1
k
i=1 p(Si ) B
qi |P | −
k−1
and
j =1
k
i=j +1 p(Si ) 2B
i=1
j =1
k
p(Si ).
i=j +1
to obtain
2qj
j =1
i=1
and therefore 3
k
qi |P |,
i=1
and we conclude that at least |P |/3 nodes of P are covered.
We continue with the proof of the correctness of the main algorithm Broadcast, which follows closely Marathe et al. [22]. Lemma 4. The number of phases of Algorithm Broadcast is O(log n). Proof. Let pi be the number of representatives before the ith phase, let qi be the number of representatives covered in the ith phase and ri be the number of roots of branches returned by the setCoverage procedure in phase i, where we exclude the branch rooted at r. Then the previous lemma gives qi pi /3, and since each branch except possibly the one rooted at r covers at least two representatives, we have ri qi /2. Therefore we have pi+1 pi − qi + ri pi − 21 qi 56 pi . Thus in each phase |P | decreases by a fraction of 1/6, and thus the number of phases is O(log n) as |P | is n − 1 to start with. The next two lemmas are the immediate counterpart of lemmas of [22] and we include their proofs for completeness. Lemma 5. The depth of the solution produced by Broadcast is O(d log n). Proof. It follows immediately by induction that after phase i, for every v ∈ V , Q contains a path of length at most d · i from some vertex of P to v. Lemma 6. The power of the output is O(B log n). Proof. The lemma follows immediately from the facts that there are O(log n) phases and in each we incur no more than 2B power. We now combine the above two lemmas to state the central result of this section:
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Theorem 7. Assuming the Min-Power Bounded-Hops Broadcast instance has a solution with power at most B, the solution produced by Algorithm Broadcast has power O(B log n) and all the nodes are reachable from the root within O(d log n) hops. 3. Min-power bounded-hops strong connectivity In this section we use the result and methods of the previous section to give an approximation algorithm for the Min-Power Bounded-Hops Strong Connectivity problem with asymmetric power requirements. We again show how to solve a budgeted version of the problem. Let OPT be an optimal solution connecting all required nodes with paths of at most d hops and assume p(OPT) B. Let v be an arbitrary vertex. OPT contains an outgoing arborescence of depth at most d, called Aout , rooted at v (so p(Aout ) B) and an incoming arborescence of depth at most d, called Ain , rooted at v (so p(Ain ) OPT). Our algorithm also computes, and then puts together, an outgoing and an incoming arborescence rooted at v. The broadcast algorithm in the previous section produces an outgoing Qout , an arborescence of depth O(d log n) rooted at v with p(Qout ) O(B log n). To obtain an incoming arborescence Qin of depth O(d log n) and with p(Qin ) O(B log n) we use almost the same algorithm, but computing incoming branches instead of outgoing branches, and a modification described below. The main Algorithm Broadcast and the setCoverage algorithm are the same, except that incoming branches are used. Incoming branches have the same power as cost (each vertex has outdegree at most 1) and this makes finding good valid branches easier. An incoming arborescence can be also found by algorithms for SLST in directed graphs. Though we are not aware of published algorithms for SLST in directed graphs, the method of [22] can be modified to handle directed graphs (as we are dealing with spanning, and not Steiner trees). Our approach, which uses spiders and set coverage instead of matching (as in [22]) also works, and we describe it below for completeness’ sake. The modified Greedy procedure, which is described below as Algorithm 4, is simpler: we only consider minimum cost path of at most d hops from the uncovered nodes to the root of the branch. Algorithm 4 Modified_Greedy(U ) Require: Set of uncovered nodes U. Implicit input parameters are the directed graph G(V , E, c), diameter bound d, budget B, and root r. Ensure: Outputs a valid incoming branch S such that p(S)/|VS ∩U | B/|U |, provided the Min-Power BoundedHops Strong Connectivity instance has a solution with power at most B; may exit with “failure” if no such solution exists. 1: for all v ∈ V do 2: EW ← ∅ 3: VW ← {v} 4: while [p(W )/|U ∩ VW | > B/|U | or (|U ∩ VW | 1 and v = r)] and [U VW ] do 5: P ← minu∈U \VW shortestBoundedPath(u, v, d) 6: EW ← EW ∪ EP , VW ← VW ∪ VP 7: end while 8: if p(W )/|U ∩ VW |B/|U | then 9: return (v, VW , EW ) 10: end if 11: end for 12: return failure We proceed with translating Lemma 2, It uses the same proof idea and is simpler. Lemma 8. Assuming the Min-Power Bounded-Hops Strong Connectivity instance has a solution with power at most B, Algorithm 4 finds a good valid branch.
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Proof. Let T be the optimum arborescence incoming into the root (which has depth d, and we assume T has power at most B) and U be the set of uncovered nodes. Traverse T in postorder and whenever a vertex v is the ancestor of at least two vertices of U (where by default every vertex is an ancestor of itself) define a branch with root v given by the subtree of T rooted at v. Remove v and its descendants from T, and repeat. The process stops when |V (T ) ∩ U | < 2. If |V (T ) ∩ U | = 1, one last branch is given by the root r and the current T. Note that every obtained branch is valid: it either covers two vertices of U, or it has r as a root. Moreover every branch obtained has depth at most d. Let Si , for 1i q, be the branches so obtained. It is immediate that p(S1 ) + p(S2 ) + · · · + p(Sq ) B. We have that |U ∩ VS1 | + |U ∩ VS2 | + · · · + |U ∩ VSq | = |U |. Therefore there is d-bounded branch (which is a spider) of optimum S with p(S)/|U ∩ VS | B/|U |, and either rS = r or |U ∩ VS | = 2. Let Q1 , Q2 , . . . , Qq , where q = |U ∩ VS |, be the paths of S from the leafs S which are in U to rS (where we include the path with zero edges from rS to itself if rS ∈ U ). For i = 1, 2, . . . , q, note that the paths Qi are edge assume disjoint and that Qi ∩ U consists of exactly one vertex, the first point of Qi , which we denote by ui , Moreover, q p(Q1 ) p(Q2 ) · · · p(Qq ). Note that for a directed path, its cost equals its power. Then p(S) = i=1 p(Qi ). During the execution of Algorithm 4, the case v = rS is considered. Let k = 1 if v ∈ U ; otherwise let k = 0. Let P1 , P2 , . . . , Pq−k be the first q − k results of the minu∈U \VW shortestBoundedPath(u, v, d) procedure and note that each Pi contains only one vertex from U \W —its first vertex—as otherwise just a part of Pi would be returned by minu∈U \VW shortestBoundedPath(u, v, d). For i = 1, 2, . . . , q − k, we have that p(Pi ) = c(Pi ) p(Qk+i ) = c(Qk+i ). Indeed, before searching for the ith path, VW does not contain all of {u1 , u2 , . . . , uk+i }. Therefore, one of Q1 , Q2 , . . . , Qk+i is a candidate to be returned by minu∈U \VW shortestBoundedPath(u, v, d). where we note that each Qi has at most d edges. If the algorithm does not find a good valid branch earlier, after q − k iterations of the while loop of Algorithm 4 (with v = rS ), W satisfies q−k q p(W ) i=1 p(Pi ) i=1 p(Qi ) = p(S) and |VW ∩ U | |VS ∩ U | and either |VW ∩ U | 2 or v = r or both. We conclude that the algorithm always finds a good valid incoming branch. Lemmas 3–6 translate immediately, with exactly the same proofs. Since p(Qout ∪Qin ) p(Qout )+p(Qin ) O(B log n), and Qout ∪ Qin is a strongly connected spanning subgraph with diameter O(d log n), we have: Theorem 9. There is a polynomial-time algorithm for Min-Power Bounded-Hops Strong Connectivity whose output has power O(p(OPT) log n) and every node is reachable from another node by a path of length O(d log n), where OPT is an optimum solution of diameter d. 4. Approximation for min-power bounded-hops broadcast in the Euclidean case The approximation factors for the broadcast problem can be improved in the case when nodes represent points on a plane, and the cost c(uv) = c(v, u) = u, v , where u, v is the Euclidean distance and is a constant. The algorithm simply shortcuts paths of O(log n) hops (as illustrated in Fig. 5) and the pseudocode of the recursive procedure is given r
log n
log n q Fig. 5. Adding shortcuts to reduce depth.
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as Algorithm 5. The procedure is initially invoked with h = d. In the algorithm, q is such that the output of the Algorithm Broadcast has depth qd. Algorithm 5 Shortcut(T , h) Require: The input is an arborescence T rooted at r with depth h · q Ensure: The output T is an arborescence rooted at r, spanning V (T ), with depth h and p(T ) p(T )q −1 . if h = 0 then return end if L ← {u : depth(u) = q} M ← {u : depth(u) q} for all ui ∈ L do Ti ← subtree of T rooted at ui Ti ← Shortcut(Ti , h − 1) end for T ← ∪ui ∈L Ti ∪u∈M\{r} {ru}. return T Lemma 10. Algorithm 5 is correct, that is, T has depth h, and p(T ) p(T )q −1 . Proof. The depth property follows immediately by induction on h. We show that pT (r)q −1 u∈M\L pT (u) and the rest follows by induction. Let u be such that c(ru)=maxv∈M c(rv) and let P = x0 = r, x1 , . . . , xj = u be the directed path in T from r to u, and note that j q. Then ru j −1 j −1 i=0 xi xi+1 , as the worst case occurs when x0 x1 = x1 x2 = · · · = xj −1 xj . −1 Note that for i = 0, 1, . . . , j − 1 we have pT (xi ) xi xi+1 and that xi ∈ M\L. Therefore pT (r) q u∈M\L pT (u) and the lemma follows. We know by Lemmas 5 and 6 that we can assume that the output of the algorithm of Section 2 has depth d · q, with q = O(log n), and power O(p(OPT) log n), where OPT is an optimum solution of depth d. Based on the discussion above we have: Theorem 11. There is a O((log n) )-approximation algorithm for Min-Power Bounded-Hops Broadcast in the Euclidean case. The same postprocessing trick cannot be directly applied to Min-Power Bounded-Hops Strong Connectivity. For the outgoing arborescence (as in Section 3), the algorithm above for Broadcast is used. For the incoming arborescence, the method is described below. If d=O(log n), then the reduction to Directed Steiner Tree gives a O((log n)2 ) approximation. Else we let q = O(log n) be a parameter given by the upper bound on the number of phases of Algorithm 1 given in Lemma 4. Every bounded-depth path chosen by Algorithm 4 is compressed immediately by adding direct edges in between nodes q hops apart in the path. The power of the path is blown up by q · q −1 . Indeed, using the argument j of the broadcast case above, for any vertex u on the path, the new power of u satisfies p (u)q −1 i=1 p(ui ), where u is the ith vertex starting with u1 = u on the path from u to the root, and j q. Thus the total new power is i −1 j p(u ) q p(u), as each p(u) appears at most q times in the middle summation. i u p (u)q u i=1 There are O(log n) phases and in each phase the power used is bounded by O(Bq ) = O(Blog n), where B is the guess for the power of the optimum solution. Using 1, we conclude: Theorem 12. Min-Power Bounded-Hops Strong Connectivity in the Euclidean case admits a O((log n)+1 )approximation algorithm.
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5. Conclusions and open problems We provided (O(log n), O(log n)) bicriteria approximation algorithms for Min-Power Bounded-Hops Broadcast and Min-Power Bounded-Hops Strong Connectivity with asymmetric cost function. In the Euclidean case, postprocessing the output of the bicriteria Min-Power Bounded-Hops Broadcast leads to a O((log n) )-approximation algorithm. We believe our approach also works for Min-Power Bounded-Hops Symmetric Connectivity with asymmetric cost function (the Krumke et al. algorithm [19] relies on the cost function being symmetric), provided that shortest BoundedPath(y, z, i) and Algorithm 3 are modified appropriately. A symmetricBoundedPath(x, y, z, i) procedure, with four parameters (vertices x, y, z, and integer i) is used to find a minimum power directed graph P which contains the edge yx and a bidirected path from y to z using at most i edges. Such a directed graph P is found by applying the method described in [28,12,24] for bounded-hops minimum cost directed paths to a slight modification of the auxiliary directed graph whose construction is given in [4] for Min-Power Symmetric Unicast with asymmetric cost function. We can obtain a polynomial-time algorithm for Min-Power Bounded-Hops Symmetric Connectivity whose output has power O(p(OPT) log n) and diameter O(d log n), where OPT is an optimum solution of diameter d. We believe the equivalent of Theorem 12 also holds for Min-Power Bounded-Hops Symmetric Connectivity in the Euclidean case. The Euclidean case might be easier to solve. However, it would be reasonable to start with SLST first, before moving to minimizing power. As far as we know, nothing better than (O(log n), O(log n)) is known for SLST when hops are used for measuring diameter and simple Euclidean distance is used for measuring cost, except for the type of “jumps” we use in Section 4 to transform bicriteria results into pure approximation algorithms. We also note that with asymmetric costs, Min-Power Multicast (even with unbounded hops) is harder than Min-Power Broadcast: [3] presents a straightforward reduction from Directed Steiner Tree to Min-Power Multicast. The recent paper of Halperin and Krauthgamer [11] shows that Directed Steiner Tree does not admit any O(log2− n) approximation unless NP has quasi-polynomial Las Vegas algorithms. The techniques seem to easily generalize to bounded degree structures, just by considering bounded degree branches (or spiders) instead of bounded depth branches in the greedy algorithm of Section 2.3. However, in the Euclidean case there is no need to specially search for bounded-degree trees, as the minimum spanning tree is within a constant optimum for all three Min-Power problems, and can be assumed to have degree at most five. Acknowledgments The presentation of the paper benefitted from the careful reading of the referees. References [1] C. Ambuehl, A.E. Clementi, M.D. Ianni, N. Lev-Tov, A. Monti, D. Peleg, G. Rossi, R. Silvestri, Efficient algorithms for low-energy bounded-hop broadcast in ad hoc wireless networks, in: STACS, 2004. [2] P. Beir, P. Sanders, N. Sivadasan, Energy optimal routing in radio networks using geometric data structrures, ICALP, Lecture notes in Computer Science, vol. 2380, 2002, pp. 366–376. [3] G. Calinescu, S. Kapoor, A. Olshevsky, A. Zelikovsky, Network lifetime and power assignment in ad hoc wireless networks, in: Proceedings of the 11th European Symposium on Algorithms, 2003. [4] G. Calinescu, I. Mandoiu, A. Zelikovsky, Symmetric connectivity with minimum power consumption in radio networks, in: Second IFIP International Conference on Theoretical Computer Science, 2002, pp. 119–130. [5] I. Caragiannis, C. Kaklamanis, P. Kanellopoulos, New results for energy-efficient broadcasting in wireless networks, in: ISAAC’ 2002, 2002. [6] M. Charikar, C. Chekuri, T. Cheung, Z. Dai, A. Goel, S. Guha, M. Li, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1) (1999) 73–91. [7] A. Clementi, A. Ferreira, P. Penna, S. Perennes, R. Silvestri, The minimum range assignment problem on linear radio networks, ESA ‘00, Lecture Notes in Computer Science, vol. 1879, 2000, pp. 143–154. [8] A. Clementi, G. Huiban, P. Penna, G. Rossi, Y. Verhoeven, Some recent theoretical advances and open questions on energy consumption in ad hoc wireless networks, in: ARACNE, 2002. [9] A.E. Clementi, M.D. Ianni, R. Silvestri, The minimum broadcast range assignment problem on linear multi-hop networks, Theoret. Comput. Sci. 1-3 (299) (2003) 751–761. [10] A.E. Clementi, P. Penna, R. Silvestri, On the power assignment problem in radio networks, Electronic Colloquium on Computational Complexity Report TR00-054. [11] E. Halperin, R. Krauthgamer, Polylogarithmic inapproximability, in: Proceedings of the 35th ACM Symposium on Theory of Computing, 2003, pp. 585–594.
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